analysis of multi-species ecological and evolutionary dynamics ecole normale supérieure, paris...
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ANALYSIS OF MULTI-SPECIES ECOLOGICALAND EVOLUTIONARY DYNAMICS
Ecole Normale Supérieure, ParisDecember 9-13, 2013
Simple models of competition and mutualism (F. Dercole)The Lotka-Volterra competition model. Symmetric vs asymmetric competition. Equilibria and isoclines. The principle of competitive exclusion. Transcritical bifurcations. A simple model of mutualism. Obligate vs non-obligate mutualism. Equilibria and isoclines. Saddle-node bifurcation.Further readingsEncyclopedia of Theoretical Ecology, Univ. California Press, 2012, pp. 88-95Proc. Roy. Soc. Lond. B (2002) 269:773-780
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The Lotka-Volterra competition modelCompetition within one population (the logistic model)
Competition within two populations
is the carrying capacity
is the intrinsic (or initial) per-capita growth rate
is the per-capita competition mortality
(adimensional) competition coefficients
symmetric competition
asymmetric competition favoring population 2 / 1
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Competition within two populations
Equilibria and isoclinesequilibria : and
the curves in the state planewhere and
isoclines :
the direction of trajectories :
the principle of competitive exclusion(Hardin G., Science 131, 1960; Gause G.F., Williams&Wilkins, 1934)
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Transcritical bifurcations (see f.r. 1)
geometric view: collision of two equilibria, as a parameter is varied, which “exchange stability”
algebraic view : a zero eigenvalue in the system’s Jacobian
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Four possible scenarios (state portraits)
coexistence dominance-2 dominance-1 mutual exclusion
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Back to the principle of competitive exclusion, consider the case ofsymmetric competition with
Mutual exclusion is the resulting scenario when competition is sufficiently strong
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A simple model of mutualismTwo species, e.g. flowers and pollinating insects, with densities and
There is intra-specific competition for commodities, as well as for other resources
The mutualism is obligate
A simple model (see f.r. 2)
where and are nonnegative increasing functions
and , , , , are positive constant parameters
The per-capita rates of commodities trading are inheritable phenotypes andthus is the prob. that an individual of species 2 receives a benefit from species 1 in the time intervalsimilarly for
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Equilibria and isoclines
equilibria : and
the direction of trajectories :
The evolution set geometric view: collision and disappearance of two equilibriaalgebraic view : a zero eigenvalue in the system’s Jacobian
The saddle-node bifurcation (see f.r. 1)